How to orbit previously weight painted vertices in Blender?
You are looking at this operation here :

Weight Paint mode


Go to Navigation panel on the 3d View
Go to View Section.
Turn on lock to cursor.
I suggest installing the "Enhanced 3d Cursor" add-on so you can snap the cursor to the mesh surfaces.
Now your viewport camera is locked to the location of the 3d Cursor.




In Edit Mode


Go to View Menu on your 3d View Panel
Look for "View Selected"
Now your viewport perspective/orthographic view will rotate around the selected object.